Rough Set Logic for Kleene's Three-valued Logic

Abstract

The rough set logic (RSL) was proposed by Duntsch and there have been many studies on the logical system based on RSL. Some researches around RSL study the deductive system for the RSL. However, the characteristics of the deductive system have not been investigated enough. In this paper, we proposed strong Kleene's three-valued logic with a natural deduction system called K3RSL. To describe a semantics for RSL, we introduce a semantic relation based on partial semantics. To handle uncertainty interpretation in RSL, it is useful to apply partial semantic relation for a meaning of deduction. This approach has been studied in many-valued logic to handle undefined and uncertain information. RSL have proposed two types of negation one is pseudocomplement and the other is dual pseudocomplement. By restriction of one negation, we can construct Kleene's threevalued logic. Application of partial semantic relation is possible to generalize the semantic interpretation of RSL. We also proposed the sketch of soundness and completeness proof for this logical system.